The Existence of a Solution to the Multi-Index Problem

The multi-index problem can be described as minimizing the cost of moving a set of p different commodities (k = 1, 2,..., p) from n origins (i = 1, 2,..., n) to m destinations (j = 1, 2,..., m). The equations then give rise to the conditions on the amount of the various types of combination that is available and required. Alternatively, the same set of restrictions arise when a single commodity has to be moved by different methods, e.g. road, rail, sea, canal, air, etc. Similarly the use of intermediate depots may require the use of a multi-index formulation. A third special type of problem where the method can be used is the capacitated transportation problem (each variable has an upper bound).